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Theorem fmptapd 7164
Description: Append an additional value to a function. (Contributed by Thierry Arnoux, 3-Jan-2017.) (Revised by AV, 10-Aug-2024.)
Hypotheses
Ref Expression
fmptapd.a (𝜑𝐴𝑉)
fmptapd.b (𝜑𝐵𝑊)
fmptapd.s (𝜑 → (𝑅 ∪ {𝐴}) = 𝑆)
fmptapd.c ((𝜑𝑥 = 𝐴) → 𝐶 = 𝐵)
Assertion
Ref Expression
fmptapd (𝜑 → ((𝑥𝑅𝐶) ∪ {⟨𝐴, 𝐵⟩}) = (𝑥𝑆𝐶))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅   𝑥,𝑆   𝜑,𝑥
Allowed substitution hints:   𝐶(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem fmptapd
StepHypRef Expression
1 fmptapd.c . . . 4 ((𝜑𝑥 = 𝐴) → 𝐶 = 𝐵)
2 fmptapd.a . . . 4 (𝜑𝐴𝑉)
3 fmptapd.b . . . 4 (𝜑𝐵𝑊)
41, 2, 3fmptsnd 7162 . . 3 (𝜑 → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐶))
54uneq2d 4158 . 2 (𝜑 → ((𝑥𝑅𝐶) ∪ {⟨𝐴, 𝐵⟩}) = ((𝑥𝑅𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶)))
6 mptun 6689 . . 3 (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = ((𝑥𝑅𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶))
76a1i 11 . 2 (𝜑 → (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = ((𝑥𝑅𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶)))
8 fmptapd.s . . 3 (𝜑 → (𝑅 ∪ {𝐴}) = 𝑆)
98mpteq1d 5236 . 2 (𝜑 → (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = (𝑥𝑆𝐶))
105, 7, 93eqtr2d 2772 1 (𝜑 → ((𝑥𝑅𝐶) ∪ {⟨𝐴, 𝐵⟩}) = (𝑥𝑆𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  cun 3941  {csn 4623  cop 4629  cmpt 5224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-opab 5204  df-mpt 5225
This theorem is referenced by:  fmptpr  7165  poimirlem3  37003  poimirlem4  37004  poimirlem16  37016  poimirlem17  37017  poimirlem19  37019  poimirlem20  37020
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